An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 299
... distinct square roots . More precise informa- tion is given by our next result . THEOREM 10.2.8 . An nxn matrix with distinct characteristic roots possesses precisely 2n or 2n - 1 distinct square roots according as it is non - singular ...
... distinct square roots . More precise informa- tion is given by our next result . THEOREM 10.2.8 . An nxn matrix with distinct characteristic roots possesses precisely 2n or 2n - 1 distinct square roots according as it is non - singular ...
Page 324
... distinct characteristic roots w1 , ... , wn . If ( t ) is any function , let ( A ) be defined by the equation ( A ) ... distinct zeros . Solve the equations = ( i ) X2 = X ; ( ii ) X + 4X + 31 = 0 . 28. A and B are matrices such that ( i ) ...
... distinct characteristic roots w1 , ... , wn . If ( t ) is any function , let ( A ) be defined by the equation ( A ) ... distinct zeros . Solve the equations = ( i ) X2 = X ; ( ii ) X + 4X + 31 = 0 . 28. A and B are matrices such that ( i ) ...
Page 325
... distinct linear factors , is similar to a diagonal matrix . 35. Let w1 , ... , we be the distinct values of the characteristic roots of A , and let , for 1 < i < k , U¿ denote the space of vectors x such that Ax = Wix . Show that , if ...
... distinct linear factors , is similar to a diagonal matrix . 35. Let w1 , ... , we be the distinct values of the characteristic roots of A , and let , for 1 < i < k , U¿ denote the space of vectors x such that Ax = Wix . Show that , if ...
Table des matières
PART | 3 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
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Expressions et termes fréquents
A₁ algebra assertion automorphism B₁ basis bilinear form bilinear operator C₁ canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix dimensionality E-operations equal equivalence EXERCISE exists follows functions given Hence hermitian form hermitian matrix identity implies inequality integers isomorphic linear combination linear equations linear manifold linear transformation linearly independent matrix group matrix of order minimum polynomial multiplication non-singular linear transformation non-singular matrix non-zero nxn matrix obtain orthogonal matrix positive definite possesses proof of Theorem prove quadratic form quadric rank reduces relation represented respect result rotation S-¹AS satisfies scalar Show similar singular skew-symmetric matrix solution square matrix suppose symmetric matrix t₁ tion triangular unique unit element unitary matrix values vector space view of Theorem w₁ write x₁ y₁ zero α₁